Elliptic curves with 2-torsion contained in the 3-torsion field

Abstract

Abstract. There is a modular curve X ′ (6) of level 6 defined over Q whose Q-rational points correspond to j-invariants of elliptic curves E over Q that satisfy Q(E[2]) ⊆ Q (E[3]). In this note we characterize the j-invariants of elliptic curves with this property by exhibiting an explicit model of X ′ (6). Our motivation is two-fold: on the one hand, X ′ (6) belongs to the list of modular curves which parametrize non-Serre curves (and is not well-known), and on the other hand, X ′ (6)(Q) gives an infinite family of examples of elliptic curves with non-abelian "entanglement fields," which is relevant to the systematic study of correction factors of various conjectural constants for elliptic curves over Q.

Elliptic curves with missing Frobenius traces

“…On the other hand, for E 6 and E 28 , the congruence obstruction is caused by nontrivial entanglements (e.g. Q(E 6 [2]) ∩ Q(E 6 [3]) = Q); the main contribution of the present paper lies in classifying the remaining (genus 0) cases where C E,r = 0 due to composite level congruence obstructions, and in clarifying the role of entanglements in those congruence obstructions.…”

“…In contrast, analogues of Conjecture 1.1 have been proven in the function field case (see [27], which also gives the statement of Conjecture 1.1 over a general number field). The best known upper bounds for π E,r (X) are π E,0 (X) ≪ X 3/4 (log X) 1/2 assuming GRH (see [46], which builds upon [38] and [33]), X 3/4 unconditionally (see [19]), and, for r = 0, π E,r (X) ≪ X 4/5 (log X) 3/5 assuming GRH, (see [46], [38] and [33]) X(log log X) 2 (log X) 2 unconditionally (see [43]).…”

“…The finite sequence (a p (E 28 ) mod 28 : p ≤ 580 and p ∤ N E28 ) is equal to (0, 1,2,26,25,22,24,19,10,9,26,22,2,24,1,10,6,10,10,16,21,21,25,22,18,23,6,20,0,19,16,11,4,17,6,16,21,16,5,24,19,15,10,26,0,14,1,3,6,14,14,21,4,18,…”

An infinite family of Serre curves

“…In their paper [BJ14], Brau and Jones describe a genus 0 modular curve X (6) whose points correspond to Q -isomorphism classes of elliptic curves, whose 2 and 3-torsion fields are more entangled than predicted making them not Serre curves. The corresponding curves have the property that the field of definition of the 3-torsion points completely contained the field of definition of the 2-torsion points.…”

Generators and number fields for torsion points of a special elliptic curve

“…In [1] Bandini and Paladino determine the number fields generated by the 3-torsion points, degrees and Galois groups of an elliptic curve y 2 ¼ x 3 þ c where c ∈ ℚ*. In [2] the result of Brau and Jones says that the rational points on the modular Fields of a special elliptic curve…”